Question

Find the eccentricity, coordinates of foci, equation of directrix and length of the latus-rectum of the hyperbola $4x^2-3y^2=36$

Answer 1

Equation of hyperbola $4x^2-3y^2=36$

$\Rightarrow \frac{x^2}{9}-\frac{y^2}{12}=1$

$\therefore a^2=9\text{and}{b}^2=12$

For eccentricity

$\because e=\sqrt{1+\frac{b^2}{a^2}}$

$\Rightarrow e=\sqrt{1+\frac{12}{9}}$

$\Rightarrow e=\frac{\sqrt{21}}{3}$

For coordinate of focus

$\because a^2=9\Rightarrow a=3$

$\therefore ae=3\times \frac{\sqrt{21}}{3}$

$\Rightarrow ae=\sqrt{21}$

$\because S(ae,0)\text{and}S\text{'}(-ae,0)$

$\Rightarrow S(\sqrt{21},0)\text{and}S\text{'}(-\sqrt{21},0)$

Equation of directrix
$x=\pm \frac{a}{e}$

$\Rightarrow x=\pm \frac{3}{\left(\frac{\sqrt{21}}{3}\right)}$

$\Rightarrow x=\pm \frac{9}{\sqrt{21}}$


Length of latus-rectum

$L.R.=\frac{2b^2}{a}$

$\Rightarrow L.R.=\frac{2\times 12}{3}$

$\Rightarrow L.R.=8$

Final answer:

$e=\frac{\sqrt{21}}{3}$, Focus $(\pm \sqrt{21},0)$, Equation of directrix $x=\mp \frac{9}{\sqrt{21}}$ and $L.R.=8$